2019 AMC 10B Problems/Problem 16
In with a right angle at , point lies in the interior of and point lies in the interior of so that and the ratio . What is the ratio
Without loss of generality, let and . Let and . As and are isosceles, and . Then , so is a 3-4-5 triangle with .
Then , and is a 1-2- triangle.
On isosceles triangles and , drop altitudes from and onto ; denote the feet of these altitudes by and respectively. Then by AAA similarity, so we get that , and . Similarly we get , and .